Abstract not found

The method of cyclic projections is a powerful tool for solving convex feasibility problems in Hilbert space. Although in many applications, in particular in the field of image reconstruction (electron microscopy, computed tomography), the convex constraint sets do not necessarily intersect, the method of cyclic projections is still employed. Results on the behaviour of the algorithm for this general case are improved, unified, and reviewed. The analysis relies on key concepts from convex analysis and the theory of nonexpansive mappings. The notion of the angle of a tuple of subspaces is introduced. New linear convergence results follow for the case when the constraint sets are closed subspaces whose orthogonal complements have a closed sum; this holds, in particular, for hyperplanes or in Euclidean space. 1991 M.R. Subject Classification. Primary 47H09, 49M45, 65-02, 65J05, 90C25; Secondary 26B25, 41A65, 46C99, 46N10, 47N10, 52A05, 52A41, 65F10, 65K05, 90C90, 92C55. Key words and p...

Abstract: We proved the existence of fixed points of non-expansive operators defined on weakly Cauchy spaces in which parallelogram law holds, the given normed space is not necessarily be uniformly convex Banach space or Hilbert space, we reduced the completeness and the uniform convexity assumptions which imposed on the given normed space.

Abstract. Stability is an important property of machine learning algorithms. Stability in clustering may be related to clustering quality or ensemble diversity, and therefore used in several ways to achieve a deeper understanding or better confidence in bioinformatic data analysis. In the specific field of fuzzy biclustering, stability can be analyzed by porting the definition of existing stability indexes to a fuzzy setting, and then adapting them to the biclustering problem. This paper presents work done in this direction, by selecting some representative stability indexes and experimentally verifying and comparing their properties. Experimental results are presented that indicate both a general agreement and some differences among the selected methods. 1

Abstract. In this paper, we prove a Halpern-type strong convergence theorem for nonexpansive mappings in a Banach space whose norm is uniformly Gâteaux differentiable. Also, we discuss the sufficient and necessary condition about this theorem. This is a partial answer of the problem raised by Reich in 1983. 1.

Abstract. The purpose of this paper is to improve [9, Theorem 3.3] by removing the hypothesis of uniform convexity. We prove the following theorem: Let X be a reflexive Banach space which has the sequentially weakly continuous duality map with gauge J. Let C be a closed convex subset of X, let Γ = {T(t) : t ≥ 0} be a strongly continuous semigroup of nonexpansive mappings on C. For fixed point u ∈ C, define the sequence {xn}n≥1 by xn = αnu+(1 − αn)T(tn)xn; for n ≥ 1, where {αn} and {tn} are real sequences satisfaying lim n→ ∞ tn tn = lim n→ ∞ αn = 0. Then the sequence {xn} converges strongly to a point of FixΓ. 1.

Abstract. In this paper, we discuss characterizations of common fixed points of commutative semigroups of nonexpansive mappings. We next prove convergence theorems to a common fixed point. We finally discuss nonexpansive retractions onto the set of common fixed points. In our discussion, we may not assume the strict convexity of the Banach space. 1.

Abstract. In this paper, using Kronecker’s theorem, we discuss the set of common fixed points of an n-parameter continuous semigroup {T(p) : p ∈ R n +} of mappings. We also discuss convergence theorems to a common fixed point of an n-parameter nonexpansive semigroup {T(p) : p ∈ R n +}. 1.

Abstract. In this short paper, we prove fixed point theorems for nonexpansive mappings whose domains are unbounded subsets of Banach spaces. These theorems are generalizations of Penot’s result in [Proc. Amer. Math. Soc., 131 (2003), 2371–2377]. 1.

Abstract. In this paper, we prove the following theorem: Let {T(t) : t ≥ 0} be a one-parameter continuous semigroup of mappings on a subset C of a Banach space E. The set of fixed points of T(t) is denoted by F ( T(t) ) for each t ≥ 0. Then F ( T(t) ) = F ( T(1) ) ∩ F ( T ( √ 2)) t≥0 holds. Using this theorem, we discuss convergence theorems to a common fixed point of {T(t) : t ≥ 0}. 1.